Optimal. Leaf size=84 \[ \frac {\left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{6 b^2 (p+1)}-\frac {a \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^2 (2 p+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1356, 266, 43} \[ \frac {\left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{6 b^2 (p+1)}-\frac {a \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^2 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1356
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx &=\left (\left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \int x^5 \left (1+\frac {b x^3}{a}\right )^{2 p} \, dx\\ &=\frac {1}{3} \left (\left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \operatorname {Subst}\left (\int x \left (1+\frac {b x}{a}\right )^{2 p} \, dx,x,x^3\right )\\ &=\frac {1}{3} \left (\left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \operatorname {Subst}\left (\int \left (-\frac {a \left (1+\frac {b x}{a}\right )^{2 p}}{b}+\frac {a \left (1+\frac {b x}{a}\right )^{1+2 p}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b^2 (1+2 p)}+\frac {\left (a+b x^3\right )^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p}{6 b^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.61 \[ \frac {\left (a+b x^3\right ) \left (\left (a+b x^3\right )^2\right )^p \left (b (2 p+1) x^3-a\right )}{6 b^2 (p+1) (2 p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 70, normalized size = 0.83 \[ \frac {{\left ({\left (2 \, b^{2} p + b^{2}\right )} x^{6} + 2 \, a b p x^{3} - a^{2}\right )} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{6 \, {\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 132, normalized size = 1.57 \[ \frac {2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} b^{2} p x^{6} + {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} b^{2} x^{6} + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} a b p x^{3} - {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} a^{2}}{6 \, {\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.71 \[ -\frac {\left (-2 x^{3} p b -b \,x^{3}+a \right ) \left (b \,x^{3}+a \right ) \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p}}{6 \left (2 p^{2}+3 p +1\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 54, normalized size = 0.64 \[ \frac {{\left (b^{2} {\left (2 \, p + 1\right )} x^{6} + 2 \, a b p x^{3} - a^{2}\right )} {\left (b x^{3} + a\right )}^{2 \, p}}{6 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 85, normalized size = 1.01 \[ {\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^p\,\left (\frac {x^6\,\left (2\,p+1\right )}{6\,\left (2\,p^2+3\,p+1\right )}-\frac {a^2}{6\,b^2\,\left (2\,p^2+3\,p+1\right )}+\frac {a\,p\,x^3}{3\,b\,\left (2\,p^2+3\,p+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {x^{6} \left (a^{2}\right )^{p}}{6} & \text {for}\: b = 0 \\\frac {a \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {a \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac {2 a \log {\relax (2 )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {a}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {b x^{3} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac {b x^{3} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac {2 b x^{3} \log {\relax (2 )}}{3 a b^{2} + 3 b^{3} x^{3}} & \text {for}\: p = -1 \\\int \frac {x^{5}}{\sqrt {\left (a + b x^{3}\right )^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\- \frac {a^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{12 b^{2} p^{2} + 18 b^{2} p + 6 b^{2}} + \frac {2 a b p x^{3} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{12 b^{2} p^{2} + 18 b^{2} p + 6 b^{2}} + \frac {2 b^{2} p x^{6} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{12 b^{2} p^{2} + 18 b^{2} p + 6 b^{2}} + \frac {b^{2} x^{6} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{12 b^{2} p^{2} + 18 b^{2} p + 6 b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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